Let \(M\)  and \(U\)  be two maximal subgroups of \(G\)  of type 2 and
index \(n\) , By [1]}, \(M_G=\operatorname{C}_G\bigl ({\operatorname{Soc}(G/M_G)}\bigr )\)  and
\(U_G=\operatorname{C}_G\bigl ({\operatorname{Soc}(G/U_G)}\bigr )\)  coincide if, and only if,
\(\operatorname{Soc}(G/M_G)\)  and \(\operatorname{Soc}(G/U_G)\)  are \(G\) -isomorphic. Hence by the
Jordan-Hölder theorem, if \(M_G\ne U_G\) , then in a given chief series of \(G\)  there exists
a chief factor \(G\) -isomorphic to \(\operatorname{Soc}(G/M_G)\)  and another chief
factor \(G\) -isomorphic to \(\operatorname{Soc}(G/U_G)\) . Consequently the number of
primitive quotients of \(G\)  associated to a maximal subgroup of
index \(n\)  is \(\operatorname{rs}_n(G)\) . By Theorem REF , the number
of core-free maximal subgroups of index \(n\)  of \(G/U_G\)  for a
maximal subgroup \(U\)  of type 2 of \(G\)  is bounded by
\(n^2\) , that is, the number of maximal subgroups of index \(n\)  of \(G\) 
with core \(U_G\)  is bounded by \(n^2\) . Hence the number of maximal subgroups of \(G\)  of type 2 and index \(n\) 
is bounded by \({\operatorname{rs}_n(G)}n^2\) .
